Mathematical model for the estrogen paradox in breast cancer treatment

Estrogen is known to stimulate the growth of breast cancer, but is also effective in treating the disease. This is referred to as the"estrogen paradox". Furthermore, short-term treatment with estrogen can successfully eliminate breast cancer, whereas long-term treatment can cause cancer recurrence. Studies highlighted clinical correlations between estrogen and the protein p53 which plays a pivotal role in breast cancer suppression. We sought to investigate how the interplay between estrogen and p53 impacts the dynamics of breast cancer, and further explore if this could be a plausible explanation for the estrogen paradox and the paradoxical tumor recurrence that results from prolonged treatment with estrogen. For this, we propose a novel ODE based mathematical model that accounts for dormant and active cancer cells, along with the estrogen hormone and the p53 protein. We analyze the model's global stability behavior using the Poincaré-Bendixson theorem and results from differential inequalities. We also perform a bifurcation analysis and carry out numerical simulations that elucidate the roles of estrogen and p53 in the estrogen paradox and its long term estrogen paradoxical effect. The mathematical and numerical analyses suggest that the apparent paradoxical role of estrogen could be the result of an interplay between estrogen and p53, and provide explicit conditions under which the paradoxical effect of long-term treatment may be prevented.